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Let $k=\Bbb F_q$ be a finite field, $\mathcal{G}$ be a reductive group over $k$, denote $m(G)$ by the minimal dimension of faithful representation of $G=\mathcal{G}(\Bbb F_q)$. Do we know the value $m(G(F_q))$ explicitly at least in the $GL_n$ case?

For example, by classification of irreducible complex representations of $GL_2(\Bbb F_q)$ (assume $q$ is not power of $2$), we know $m(GL_2(\Bbb F_q))=q-1$ (a suitable cuspidal representation of dimension $q-1$ is faithful ).

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    $\begingroup$ Have you looked at Landazuri-Seitz? Others such as Kleidman, Liebeck, Tiep, Zalesski may give relevant information $\endgroup$ – Geoff Robinson Jun 26 '18 at 16:14
  • $\begingroup$ @GeoffRobinson Thank you, Landazuri-Seitz dealt with projective representations and gave some good bounds. Maybe there are also some results for ordinary representations. $\endgroup$ – zzy Jun 27 '18 at 3:00
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    $\begingroup$ A lower bound for a projective representation is also a lower bound for an ordinary representation. $\endgroup$ – Geoff Robinson Jun 27 '18 at 19:36

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